10.2
Summation Notation
Summation notation is used to write sums of terms that follow a pattern.
The Greek letter sigma stands for summation. It signals to add values from a starting term to an ending term.
The number below the sigma shows where to start summing, and the number above tells where to stop. A summation connects directly to sequences, since each sum is formed by adding consecutive terms of a sequence.
Each term in the sum is formed by plugging the index value into a given expression.
Now, imagine a homeowner tracks electricity generation from a solar panel during the first ten days of spring.
As daylight varies, daily production varies, with outputs labeled E1 through E10.
Instead of writing E1 plus E2 plus E3 and continuing to E10, the total is expressed using a compact summation.
This represents the addition of daily outputs regardless of weather variations.
Extending this summation to three hundred sixty-five days provides annual solar output.
Comparing yearly solar output with household electricity needs shows how much of the home’s energy needs are covered by solar.
Sigma notation, also known as summation notation, provides a concise method for representing the sum of a sequence of terms that follow a regular pattern. It utilizes the uppercase Greek letter sigma (∑), A typical expression is:
In this form, k the index of summation is 1, the starting value, and n the ending value. The term ak represents the general term of the sequence.
For example, the increasing sequence 5, 7, 9, ..., 23 over 10 terms can be expressed as:
This simplifies the representation of such patterns in a compact form.
Sigma notation follows useful algebraic properties that make it easier to simplify and manipulate sums:
These properties are grounded in the associative and distributive laws of real numbers, allowing for efficient evaluation and transformation of sums. As well as this, sigma notation can accommodate varying starting indices and expressions, broadening its applicability in both theoretical and applied mathematical contexts, such as evaluating series, modeling growth patterns, or calculating cumulative totals in real-world scenarios.
Summation notation is used to write sums of terms that follow a pattern.
The Greek letter sigma stands for summation. It signals to add values from a starting term to an ending term.
The number below the sigma shows where to start summing, and the number above tells where to stop. A summation connects directly to sequences, since each sum is formed by adding consecutive terms of a sequence.
Each term in the sum is formed by plugging the index value into a given expression.
Now, imagine a homeowner tracks electricity generation from a solar panel during the first ten days of spring.
As daylight varies, daily production varies, with outputs labeled E1 through E10.
Instead of writing E1 plus E2 plus E3 and continuing to E10, the total is expressed using a compact summation.
This represents the addition of daily outputs regardless of weather variations.
Extending this summation to three hundred sixty-five days provides annual solar output.
Comparing yearly solar output with household electricity needs shows how much of the home’s energy needs are covered by solar.
From Chapter 10:
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